(a) Verify that y1(x) = ex is a solution of xy '&#...

(a) Verify that y₁(x) = e^x is a solution of

xy '' − (x + 10)y’ + 10y = 0.

(b) Use (5) to find a second solution y₂(x). Use a CAS to carry out the required integration.

(c) Explain, using Corollary (a) of Theorem 3.1.2, why the second solution can be written compactly as

Reference:

Theorem 3.1.2: Superposition Principle—Homogeneous Equations

Let y₁, y₂,. . ., y_k be solutions of the homogeneous nth-order differential equation (6) on an interval I. Then the linear combination

y = c₁ y₁(x) + c₂ y₂(x) + . . . + c_k y_k(x),

where the c_i, i = 1, 2, . . . , k are arbitrary constants, is also a solution on the interval.